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    You are here: Modules > Iterations
    Links in the text refer to the lower part of the page

    Iterations

    Calculate algebraic series such as e = 1+ 1/2! + 1/3! + ..., a square wave, Fibonacci numbers. Study iterative maps, e.g. the (one-dimensional) Logistic map: (see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simples mathematical system showing many characteristics of the development of chaotic behaviour.
    Several analytical tools are valailable to study the results of Iterations and ODE's such as:      
         time series | Power spectra | 2D projections | Fixed points | Lyapunov exponents
    A special window has been added in Mathgrapher v2 to allow detailed presentation of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

    Examples:
    Logistic map: Sensitivity to initial conditions Projection in 2D Power spectrum Bifurcation diagram Lyapunov exponents
    Henon map: Definition 2D orbit Region of Stability
    Mandelbrot and Julia sets: Definition Mandelbrot: vary parameters Julia: vary initial conditions

    Chaos

    In the second half of the last century it was discovered that most (non-linear) dynamical systems exhibit choatic behaviour. This came as a surprise. It was thought that deterministic systems would behave in a predictable way and that choatic behaviour in such systems would be due to many different external influences on a system. It turned out, however, that non-linear systems which are fully deterministic and quite simple and straightforward in mathematical sense often show chaotic behaviour. This means that a very small change in the initial conditions of the system may may lead to completely different end results thereby making the outcome completely unpredictable. This was surprising because it occurs in systems whose behaviour is governed by well-understood physical principles described by exact mathematical formulae. The example that is often mentioned is the butterfly in Alaska which may cause a thunderstorm in Texas just by flapping its wings. Scientist may be surprised by this finding, however every pinball player knows that a very slight change in the velocity of the ball at some point along its trajectory (for instance by kicking the machine) may change the end result completely.

    The examples contain two well-known examples of systems that exhibit chaotic behaviour. Both are incorporated in the Demonstrations, so you just have to choose Demonstrations from Mathgrapher's menu to see how you may calculate and analyse such systems. The first example is a simple iterative algorithm: the logistic map. The second example is a well known system of Ordinary Differential Equations (ODE's): the Lorenz equations.

     
     
     
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